Revisit a SIR System: Using the Dynamical System Approach to Detect Slow-Fast Oscillations

• Xiangyuan Liang ^[1] ; Pei Yu ^[1]
  1. [1] Western University

     Western University

     Azerbaiyán

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This study investigates the dynamical behavior of a modified SIR model that incorporates a small intrinsic population growth rate. Traditional approaches, such as Geometric singular perturbation theory (GSPT), have been used to identify recurrent oscillations when the growth rate is negligible. However, these techniques often break down or become analytically intractable over broader parameter regimes. To overcome these limitations, we employ tools from nonlinear dynamical systems theory, namely bifurcation analysis, center manifold theory, and normal form analysis, to characterize periodic solutions and assess their stability. In particular, we focus on identifying Hopf bifurcations and examining whether oscillatory behavior persists beyond the narrow conditions typically required by GSPT. Building on previously proposed criteria for detecting slow–fast dynamics, we refine and extend this methodology, demonstrating that it remains effective even when the growth rate is not especially low. This extension broadens the applicability of the approach to a wider class of dynamical systems.

    Numerical simulations complement the theoretical analysis and illustrate the model’s ability to exhibit robust, recurrent epidemic cycles across a wide range of parameter values.

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